In the ENZ-approach, the object has been restricted initially
to a spatial delta-function or to a tiny transmitting hole that is small
compared to the diffraction unsharpness in object space. The exact calculation
of images of more complicated object patterns is a subject of much practical
interest. Recently, a computation scheme has been constructed in which the ENZ-formalism is used to calculate the through-focus image of a general 3D object. This scheme partly relies on electromagnetic field solvers, such as FDTD or a Finite Element Solver, which calculate the field in the entrance pupil of the imaging apparatus resulting from the interaction between the incident illumination and the object. Having available the complex field in the entrance pupil of the imaging system, one can represent this field in a Zernike expansion after which the through-focus image follows straightforwardly from ENZ-theory. This newly proposed scheme to image extended objects, such as masks in advanced lithography, is potentially very fast as it exploits an important property of the ENZ-formalism: the transition between the entrance pupil and image region involves basic function that can be calculated (and stored in a look-up table) in advance. This new approach to mask imaging shows large potential in mask optimization for advanced lithography. In this field many consecutive mask imaging operations are performed in iterative processes to optimize the wafer image. In addition to the expected reduction of calculation times, the ENZ-based scheme has another convenient property: given a general object, the ENZ-formalism calculates the resulting field in the focal volume. This means that the ENZ-formalism provides instant information about the through-focus behaviour of the image. This information is related to the depth-of-focus and therefore very valuable to the lithographic community.

In the figure below, we show a simple example of an extended object. We
use the ENZ-theory to calculate the intensity distribution in the image plane of two adjacent
point sources. Note that, with increasing coherence in the illumination, the presence
of two separate objects is more easily detected.

The intensity distribution in the image plane of a perfect optical system
for different degrees of coherence in the illumination of the object (two
adjacent point sources). By decreasing the coherence (higher sigma-value),
one observes an increased merging of the two objects

In the next example, a far more complicated 3D object is considered (see first row figure below). This object is illuminated with a plane wave after which the resulting near-field is determined using FDTD. Next, this near-field is propagated to the entrance pupil of the imaging apparatus. Subsequently, the field in the entrance pupil is represented in a Zernike expansion, after which the resulting image follows straightforwardly according to the ENZ-formalism (see bottom row figure below). In the figure we show three distinct images, corresponding to an image position before, at and after the predicted paraxial best image position.

Electron microscopy

In the case of low-NA electron microscopy, the aberrations
of an electron microscope are relatively small and comparable to the wavelength
of the electron matter waves. The ENZ-formalism thus can be applied to
electron microscope through-focus images in order to characterize the
instrument itself and/or the angular distribution of the electron radiation
emitted by the source.

Stochastic media (atmospheric turbulence)

Zernike expansion has been successfully used to describe
the stochastic properties of random media (e.g. the earth atmosphere at
visible wavelengths). Imaging through the atmosphere can be described
by ENZ-theory and the correction of the atmospheric perturbation becomes
feasible if the Zernike coefficients can be retrieved. A practical application
can be found in so-called ‘adaptive optics’ for astronomy.